Math Project-based Learning

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"What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative, than this of mathematics?" (Benjamin Franklin; American author, printer, scientist, inventor, and diplomat; 1706-1790.)


Summary

"Tell me, and I will forget. Show me, and I may remember. Involve me, and I will understand." (Confucius; Chinese thinker and social philosopher; 551-479 B.C.)

This document provides an introduction to uses of Project-based Learning (PBL) in math education. It includes arguments supporting use of PBL and it includes a number of examples that can be adapted for use in a wide range of math courses at the pre-college level and in teacher education.

Project-based Learning is a way to involve students in learning and using math. It is a good vehicle for helping students make progress on a number of math educational goals not directly covered in the "traditional" math curriculum. Not least among these goals is to help make math education a pleasant and rewarding discipline of study that adults will look back on with fond memories.

The target audience for this document is preservice and inservice K-12 teachers who teach math, and teachers of such teachers. Keep in mind that the overriding goal or purpose of this document is to improve math education. As a teacher, you should consider making use of PBL when:

  1. You believe it will help to improve the quality of the math education your students are getting; and
  2. You believe it will help you learn more about teaching math and the ways in which your students learn math.

For some general information about project-based learning see:

  • edutopia (n.d.). What works in education. George Lucas Education Foundation. Retrieved 3/2/2016 from http://www.edutopia.org.

Introduction

"A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." (G.H. Hardy; English mathematician; 1877-1947.)

In this document, the acronym PBL is used to represent project-based learning. Problem-based learning will be written as Problem-based learning. The two topics overlap, but are not the same.

PBL Venn.jpeg


The main focus in this document is on Project-based learning. There, a student or a team of students—perhaps even the whole class or whole school—works on a project over an extended period of time. The student or students have some choice (indeed, perhaps a great deal of choice) on the topic they will address. Project-based learning is oriented toward producing a product, performance, or presentation.

A math PBL lesson typically has broader goals than just learning math content that will be assessed via traditional math tests. For example, one standard goal in math PBL is to help increase student levels of Math Maturity. Quoting from the Math Maturity website:

It is well recognized that some rote memory learning is quite important in math education. However, most of this rote learning suffers from a lack of long-term retention, and from the learner’s inability to transfer this learning to new, challenging problem situations both within the discipline of math as well as to math-related problem situations outside the discipline of math.
Thus, math education has moved in the direction of placing much more emphasis on learning for understanding and for solving novel (non-routine) problems. There is substantial emphasis on learning some “big ideas” and gaining math-related "habits of mind and thinking skills" that will last a lifetime.
There is still more to achieving a useful level of math maturity. A student needs to learn how to learn math, how to self-assess his or her level of math content knowledge, skills, and math maturity, how to relearn math that has been forgotten or partially forgotten, how to make effective use of online sources of math information and instruction, and so on.

A Math Project-based Learning Example

Here is an example of math project-based learning. Students are given a general project area, and they are asked to work on it individually or in teams:

Select an academic discipline other than math. Investigate roles of math in helping to represent and solve the problems in this discipline. Pay special attention to what areas of math are important in the academic discipline you select, and why they are important. The specific assignment is to produce a paper and do an oral presentation that can help inform your fellow students.

This type of assignment could well extend over a number of weeks, with much of the required work being done outside of class. If this assignment is given in a specific math class (such as geometry or algebra) then the assignment might include the requirement that the study focus on roles of the math being studied in the class and that these be emphasized in the project.

The paper and presentation might become part of a student's overall academic portfolio or math portfolio. As a teacher in this setting, you might want to emphasize the idea of students developing a paper that will be useful to other students in the class, students in other similar classes, and future students in the class. And, of course, the work done by your students helps to represent the quality of instruction you are providing for your students. The intended audience is much larger than just the teacher!

A Math Problem-based Learning Example

This section provides some information about problem-based learning and how it differs from project-based learning. Quoting from an ERIC Digest article on Problem-based Learning in Mathematics:

Problem-Based Learning (PBL) describes a learning environment where problems drive the learning. That is, learning begins with a problem to be solved, and the problem is posed in such a way that students need to gain new knowledge before they can solve the problem. Rather than seeking a single correct answer, students interpret the problem, gather needed information, identify possible solutions, evaluate options, and present conclusions.

Here is an example of math problem-based learning:

This is the four 4s math problem. The goal is to combine four 4s in various ways in order to make as many different integers as possible. The "combine" rules are that one can use addition, subtraction, multiplication, division, and parentheses.
Thus: (4 + 4)/(4+ 4) = 1; 4/4 + 4/4 = 2; and (4 + 4 + 4)/4 = 3.
For a more complex version of the problem, also allow concatenation (thus, 44/44 = 1 and 444/4 = 111), exponentiation, or other types of operations.

This math problem and its variations are widely used in math education. It illustrates that a math problem may have more than one solution. It illustrates the need for very careful definition of a problem. It is a problem that can engage individual students or a team of students over an extended period of time. Thus, you can see it has some of the characteristics of a project. However, typically all students are required to work on the exact same problem in problem-based learning. Some students are likely to produce more answers than others.

There are many variations of the problem, both in the base number (for example, how about using four 3s) and the allowable operations. A 3/2/2016 Google search of the expression four fours math problem produced over 32 thousand results.

This Web search illustrates an important aspect of math education. If the goal in this problem is to "get an answer," then a Web search will achieve this goal. However, suppose the goal is to increase one's math problem solving, mental arithmetic, and systematic search skills. Then it is "cheating" to use the Web—much in the same way it is cheating to copy a fellow student's paper.

For more information about problem-based learning see: IAE-pedia document: Math Problem-based Learning.

A Combined Math Problem and Project

A 3/2/2016 Google search of the expression math project-based learning produced about 73 million results. A Google search of the expression math problem-based learning produced over 32 million results. Many of these hits are documents covering both project-based and problem-based learning. There is no fine dividing line between what constitutes a project and what constitutes a problem.

Consider the topic of carbon footprint, which is part of issues such as global warming and sustainability. Quoting from the linked website:

Inevitably, in going about our daily lives — commuting, sheltering our families, eating — each of us contributes to the greenhouse gas emissions that are causing climate change. Yet, there are many things each of us, as individuals, can do to reduce our carbon emissions. The choices we make in our homes, our travel, the food we eat, and what we buy and throw away all influence our carbon footprint and can help ensure a stable climate for future generations.

Very roughly speaking, there is the problem of defining and measuring carbon footprint, and there is the project of taking action to reduce the carbon footprint of a person, a factory, or a city. Defining and measuring carbon footprints are science problems, and this science certainly involves quite a bit of mathematics.

A simplified version of the results can be expressed in a carbon footprint calculator. Just follow the step-by-step set of instructions and you can get an estimate of your personal carbon footprint or your household's carbon footprint.

Hmm. (Imagine light bulbs going off in my head). The questions asked by this carbon footprint calculator give you some information about details of carbon footprint, but in no sense provide comprehensive coverage of the topic. Think of a special math calculator that calculates the hypotenuse of a right triangle from input giving the lengths of the two other sides. A person can easily follow the instructions, but gains little insight into the Pythagorean theorem. You might want to think about students memorizing the Pythagorean theorem and developing skill and accuracy in using it—with the assurance that "mathematicians have proven that it works." Once you think about this, do some similar thinking about how many students in math classes get by on a "memorize the formulas and use them with little or no understanding" approach.

Now, back to possible student carbon footprint projects. How much would it impact the carbon footprint of the students' school or their individual households if all students and household members routinely wore a sweater and the thermostat were lowered by two degrees Fahrenheit? What can students to to cause their thermostat lowering to happen in their school or in many households in their town/city?

This example and the Hmm paragraph provide a useful idea in the development and use of math projects in a math class. For each math topic being covered, there is the math to be learned and also some of the uses of the math. A good project increases a student's understanding of the uses of the math topic. It typically requires use of other math topics, and helps the integration of these topics.

Math Project-based and Problem-based Collaborative Learning

Both project-based learning and problem-based learning can be done by individuals or by teams. When done by teams, PBL is an example of collaborative learning.

Collaborative learning has long been an important aspect of schooling. Computer technology now makes it possible for the collaborators to be widely separated from each other. An important aspect of a modern education is for students to learn to collaborate in both face-to-face and long-distance modes.

There has been substantial research on collaborative learning. Here are three short sections quoted from this article by Brigid Barron and Linda Darling-Hammond:

Traditional academic approaches—those that employ narrow tasks to emphasize rote memorization or the application of simple procedures—won't develop learners who are critical thinkers or effective writers and speakers. Rather, students need to take part in complex, meaningful projects that require sustained engagement and collaboration.
Productive Collaboration. A great deal of work has been done to specify the kinds of tasks, accountability, and roles that help students collaborate well. In a summary of forty years of research on cooperative learning, Roger and David Johnson, identified five important elements of cooperation across multiple classroom models:
  • Positive interdependence
  • Individual accountability
  • Structures that promote face-to-face interaction
  • Social skills
  • Group processing
Evidence shows that inquiry-based, collaborative approaches benefit students in learning important twenty-first-century skills, such as the ability to work in teams, solve complex problems, and apply knowledge from one lesson to others. The research suggests that inquiry-based lessons and meaningful group work can be challenging to implement. They require changes in curriculum, instruction, and assessment practices—changes that are often new for teachers and students.

Project-based Learning Is a Team Activity

"Individual commitment to a group effort—that is what makes a team work, a company work, a society work, a civilization work." (Vince Lombardi; American football coach; 1913-1970.)

Project-based learning is a team learning and doing activity. Nowadays in a school setting, a project-based learning team consists of:

  1. One or more students who are in charge of the project. They may be widely dispersed in location.
  2. Other people, such as peers, siblings, parents, teachers, and so on. They serve as advisers, sources of information, and sources of formative and summative feedback.
  3. Virtual and physical libraries. Think of the collected available knowledge of the human race as a tool.
  4. Tools, to aid the physical and cognitive capabilities of the students who are doing the project.
  5. Other resources such as materials, money, facilities, and environments in which a project is being carried out or is focused on.

At first glance, this may seem like a strange way to think about membership on a project team. The idea being emphasized is that project-based learning is always done in a team environment, even if there is only one student directly involved. Even a one-person team draws upon the accumulated knowledge and skills of a huge collection of people and other resources. The Web, for example, represents the past and continuing work of many millions of people. The tools (including carpenter and computer tools) we routinely use represent the thinking and production work of a large number of people.

One of the most important goals in project-based learning is for students to learn to make effective use of these five different types of team members. It is a valuable life skill. Notice that this goal is independent of any specific content area that a project might focus on. The expertise one develops in working in this team environment readily transfers to other projects.

Another important goal is for students to gain increased confidence in their ability to accomplish complex tasks that are probably beyond their ability to accomplish without the aid of other "members" of the team. Such accomplishments help to build self-esteem.

Math Is a Large and Old Discipline

"In most sciences one generation tears down what another has built and what one has established another undoes. In mathematics alone each generation adds a new story to the old structure." (Hermann Hankel; German mathematician; 1839-1873.)
"If I have seen further it is by standing on the shoulders of giants." (Isaac Newton; English mathematician & physicist; 1642-1727.) (Note that Albert Einstein and many others have made similar statements.)

The history of math predates the first development of reading and writing that occurred more than 5,000 years ago. The development of reading, writing, and math notational systems laid the foundations for the "basics" of reading, writing, and arithmetic in our current educational system. Thus, our formal educational system has more than 5,000 years of experience in developing and teaching math in schools.

Math is a broad and deep discipline that people learn through both informal and formal education. Each academic discipline or area of study can be defined by a combination of general characteristics such as:

  • The types of problems, tasks, and activities it addresses.
  • Its accumulated accomplishments such as results, achievements, products, performances, scope, power, uses, impact on the societies of the world, ability to attract followers and supporters, and so on.
  • Its history, culture, and language, including notation and special vocabulary.
  • Its methods of teaching, learning, assessment, and thinking. What it does to preserve and sustain its work and pass it on to future generations.
  • Its tools, methodologies, and types of evidence and arguments used in solving problems, accomplishing tasks, and recording and sharing accumulated results.
  • The knowledge and skills (the levels of math expertise; the levels of math maturity) that separate and distinguish among: a) a novice; b) a person who has a personally useful level of competence; c) a reasonably competent person, employable in the discipline; d) a regional or national expert; and e) a world-class expert.

A little thought should convince you that it is not easy to carefully define a particular discipline in a manner that differentiates it from other disciplines. What do two different disciplines such as math and history have in common, and how are they different? It is appropriate to give students the following type of project assignment:

Select a non-math discipline of study. Give a good definition of this discipline, and a good definition of the discipline of math. Compare and contrast the two disciplines. In this compare and contrast, make sure you cover all of the general defining characteristics of a discipline (see the bulleted list given above). Produce a report that is suitable for presentation to the whole class and that will be turned in to your teacher.
Alternatively, such as activity can be done in small groups in a class discussion mode, with results discussed and shared with the whole class.

The Discipline of Math, and Math Project-based Learning

Math is a human endeavor. The current—and still rapidly growing—discipline of mathematics is the result of many tens of thousands of contributors over thousands of years.

Think about your math-related insights into the second item in the six-item list of defining characteristics of a discipline:

Its accumulated accomplishments such as results, achievements, products, performances, scope, power, uses, impact on the societies of the world, ability to attract followers and supporters, and so on.

What do you know about these aspects of math, and what do you want your students to know? These areas are not covered in state and national math assessments. However, they are important aspects of math as a human endeavor. They are a rich source of projects in a math course.

Think about your math-related insights into the third item in the list of defining characteristics of a discipline:

Its history, culture, and language, including notation and special vocabulary.

Math is often called a language. Learning to read, write, speak, listen, and communicate in the language of mathematics is a critical aspect of learning math. Math journaling is a very useful and ongoing individual student project that contributes to students learning to communicate in the language of mathematics.

The history of calculators and computers is closely intertwined with the overall history of math. Calculators and computers were initially developed as aids to doing arithmetic. Now, of course, computers are important aids to representing and solving problems in every academic discipline. A math-centric way to think about this is that computers represent one of the ways that math is indispensable in every academic discipline. Certainly as one studies uses of math throughout the curriculum, calculators and computers must be given substantial emphasis.

The history and applications of math provide a wide range of possible projects for project-based learning.

Finally, consider the whole list of characteristics of a discipline, and think about them in terms of math and math education. When you are asked, "What is math?" what is your response? What response might your students give? What response would you like them to give? Clearly, the discipline of math is much more than the specific math topics you teach during math classes. Project-based learning can provide opportunities for students to greatly broaden their insights into the discipline of mathematics.

Goals of Math Education

“Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems.” (George Polya; Hungarian and American math researcher and educator; 1887-1985.)

Here's a common-sense suggestion to math teachers. Make use of math project-based learning to:

  • Help meet math learning goals that are better achieved in a project-based learning environment than through other ways of teaching math. Keep in mind that some of these goals are "traditional" while others may not be currently met in a traditional math curriculum.
  • Help students learn to do projects that involve math. This includes helping students learn some roles of math in the overall multidisciplinary area of project-based learning.

There are many possible goals in math education. Our math education system has identified some of these goals and emphasizes them in math curriculum content, instructional processes, and assessment. The next four sub-sections discuss math education goals.

Standards, Such as the Common Core State Math Standards

In the United States, each state establishes its own goals (standards, benchmarks) for math education at the K-12 level. They are assisted in this endeavor by the work of the National Council of Teachers of Mathematics (NCTM) and a variety of other groups and organizations. In recent years, the Common Core State Math Standards have dominated. Quoting from the linked document:

For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on this promise, the mathematics standards are designed to address the problem of a curriculum that is “a mile wide and an inch deep.”
These new standards build on the best of high-quality math standards from states across the country. They also draw on the most important international models for mathematical practice, as well as research and input from numerous sources, including state departments of education, scholars, assessment developers, professional organizations, educators, parents and students, and members of the public.
The math standards provide clarity and specificity rather than broad general statements. They endeavor to follow the design envisioned by William Schmidt and Richard Houang (2002), by not only stressing conceptual understanding of key ideas, but also by continually returning to organizing principles such as place value and the laws of arithmetic to structure those ideas.

In addition, the “sequence of topics and performances” that is outlined in a body of math standards must respect what is already known about how students learn. Therefore, the development of the standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skills, and understanding develop over time. The knowledge and skills students need to be prepared for mathematics in college, career, and life are woven throughout the mathematics standards.

Mathematics: What should students learning and when should they learn it? was a 2007 conference exploring the ongoing challenge of what students should/could be learning in K-12 mathematics. Jere Confrey's paper presented at this conference is titled, Tracing the evolution of mathematics content standards1 in the United States: Looking back and projecting forward towards national standards. Quoting from her paper:

However, the original use of the term “standard” was that of the king’s standard; waving above the battlefield, representing the king’s authority; it was in a real sense fought for as though it were the king himself (Oxford English Dictionary, n.d.). Thus the earliest standards were essentially authoritarian—proclamations of religious and political leaders, accompanied by no explanation and no justification other than the authority as warranted by the full and unchallenged status vested in the proclaimer.
The term “standard” subsequently evolved, becoming closely woven into the emergence and subsequent maturation of fields of science and technology. As science flowered during the 1600s, there were increasing needs to coordinate results and findings. The second aspect of the term “standards” rested on coordinating measurement from one place to another so that the accuracy of results could be secured by a common metric, that is, by “a standard of measurement.” Metrics were often first shaped to fit the circumstance, sized for human convenience—the meter or yard as the length of an arm, the pace with the foot, and so on—taking into account such qualities as convenience, portability, relative size, and reliability. While human needs and preferences were expressed in the metrics, an outside world simultaneously pushed back on the measures, demanding standardization across place and time. Standardization constrains variations, and permits other discoveries of regularities and new inventions often at a higher level of understanding.

The second of these two paragraphs captures the challenge in education. It is not enough to make a statement that math is part of the required K-8 curriculum. There need to be measures (metrics) of what is studied and what it is students are expected to learn. There must be justification for the standards that are being promoted and/or required.

George Polya

George Polya.png
"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery." (George Polya; Hungarian and American math researcher and educator; 1887-1985.)


There are many other ways to approach the question of goals for math education. For example, George Polya was a world class math researcher and math educator. In about 1969 he gave a talk on The Goals of Mathematics Education. Quoting from that talk:

Now what about mathematics teaching? Mathematics in the primary schools has a good and narrow aim and that is pretty clear in the primary schools. An adult who is completely illiterate is not employable in a modern society. Everybody should be able to read and write and do some arithmetic, and perhaps a little more. Therefore the good and narrow aim of the primary school is to teach the arithmetical skills—addition, subtraction, multiplication, division, and perhaps a little more, as well as to teach fractions, percentages, rates, and perhaps even a little more. Everybody should have an idea of how to measure lengths, areas, volumes. This is a good and narrow aim of the primary schools—to transmit this knowledge—and we shouldn't forget it.
However, we have a higher aim. We wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good school of thinking. But what is thinking? The thinking that you can learn in mathematics is, for instance, to handle abstractions. Mathematics is about numbers. Numbers are an abstraction. When we solve a practical problem, then from this practical problem we must first make an abstract problem. Mathematics applies directly to abstractions. Some mathematics should enable a child at least to handle abstractions, to handle abstract structures. Structure is a fashionable word now. It is not a bad word. I am quite for it.
But I think there is one point which is even more important. Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. For the higher aims about which I am now talking are some general tactics of problems -- to have the right attitude for problems and to be able to attack all kinds of problems, not only very simple problems, which can be solved with the skills of the primary school, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. But the foundations should be started in the primary school. And so I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems.

Polya emphasized learning to think mathematically, to deal with math abstractions, and to deal with complicated problems. This type of math education is designed to help develop a student's level of math maturity. His ideas help form the basis of math problem-based and project-based learning

Math Maturity

"A growing level of math maturity is shown by 'a significant shift from learning by memorization to learning through understanding.'" (See Wikipedia attribution below.)

Quoting from the Wikipedia: "Mathematical maturity is a loose term used by mathematicians that refers to a mixture of mathematical experience and insight that cannot be directly taught, but instead comes from repeated exposure to complex mathematical concepts."

The IAE-pedia document Math Maturity includes the following list of aspects of math maturity:

  • Communicate mathematics and math ideas orally and in writing using standard notation, vocabulary, and acceptable style.
  • Learn to learn math; complete the significant shift from learning by memorization to learning through understanding.
  • Generalize from a specific example to a broad concept.
  • Progress in the development of the field of math is built on developing broad concepts—general ideas, vocabulary and notation, and proofs.
  • Transfer one’s math knowledge and skills into math-related areas and problems in disciplines outside of mathematics.
  • Use concrete references appropriately as an aid to learning, an aid to problem solving, and as an aid to help others learn math.
  • Handle abstract ideas without requiring concrete referents.
  • Manifest mathematical intuition by abandoning naive assumptions and by readily drawing on one’s accumulated subconscious math knowledge and insights. This intuition includes having a "feeling" for the correctness or incorrectness of math-related assertions within the realm of math that one has studied.
  • Move back and forth between the visual (e.g., graphs, geometric representations) and the analytical (e.g., equations, functions).
  • Recognize a valid mathematical or logical proof, and detect "sloppy" thinking. Provide solid evidence (informal and formal arguments and proofs) of the correctness of one’s efforts in solving math problems and making proofs.
  • Seek out and recognize mathematical patterns.
  • Separate key ideas from the less significant ideas in problem solving.
  • Represent (model) verbal and written problem situations as mathematical problems. (Translate "word problems" into math problems.)
  • Draw upon one’s math knowledge and skills to effectively address novel (not previously encountered) math-related problems.
  • Pose and/or recognize math problem situations of interest to oneself and others.
  • Understand the capabilities and limitations of tools (including calculators and computers) designed to help represent and solve math problems. Learn to make effective use of these tools at a level commensurate with one's overall knowledge, skills, and understanding in math.

Quoting Larry Denenberg:

Thirty percent of mathematical maturity is fearlessness in the face of symbols: the ability to read and understand notation, to introduce clear and useful notation when appropriate (and not otherwise!), and a general facility of expression in the terse—but crisp and exact language that mathematicians use to communicate ideas. Mathematics, like English, relies on a common understanding of definitions and meanings. But in mathematics definitions and meanings are much more often attached to symbols, not to words, although words are used as well. Furthermore, the definitions are much more precise and unambiguous, and are not nearly as susceptible to modification through usage. You will never see a mathematical discussion without the use of notation!

One of the most important goals in math education is to help students gain in their levels of math maturity. However, math maturity is difficult to assess, and thus is seldom assessed.

Enjoying Math

"We cannot hope that many children will learn mathematics unless we find a way to share our enjoyment and show them its beauty as well as its utility." (Mary Beth Ruskai; American mathematician and researcher; 1944-.)

Many adults have had math education experiences that lead them to say things like, "I hate (or, hated) math" and "I can't do math." Many people believe that a math education that produces the "I hate math" and related types of results is significantly flawed.

Thus, math teachers might want to place greater emphasis on helping students to enjoy and appreciate the math they are learning and the math learning experiences. Quoting Alfred Posamentier, Professor of Mathematics Education at the City College of New York, from his 2002 New York Times article.

The point is to make math intrinsically interesting to children. We should not have to sell mathematics by pointing to its usefulness in other subject areas, which, of course, is real. Love for math will not come about by trying to convince a child that it happens to be a handy tool for life; it grows when a good teacher can draw out a child's curiosity about how numbers and mathematical principles work. The very high percentage of adults who are unashamed to say that they are bad with math is a good indication of how maligned the subject is and how very little we were taught in school about the enchantment of numbers.

Information Age Education started the publication of a long sequence of articles on Joy of Learning beginning in December, 2015. The essence of this sequence is that schools should be joyful places and that there is substantial research supporting the benefits of designing learning to be a joyful experience.

Folk Math

Math is taught in schools throughout the world. However, there are many children who never attend school. Many others receive only a few years of formal education. A number of people have done studies of street-people who have learned the math they need without benefit of much (or, perhaps any) formal schooling.

Gene Maier has written excellent materials on the idea of Folk Math. Think in terms of folk music or folk art—knowledge and skills that are learned in informal settings but that meet the needs of the learners. Also, think in terms of the math that children learn as they memorize counting rhymes, or play board games, card games, and computer games even before they start school.

In terms of project-based learning, think about designing a project for students to explore two areas:

  • What math do people learn informally (on the streets) in various places throughout the world? Pay special attention to children who receive little or no formal schooling.
  • What math do students learn in school in different parts of the world? How is this the same and how is it different from the math that students are learning in the United States?

The topic of Folk Math raises other issues. What math do typical adults with a high school education or more use in their everyday lives? For example, do their everyday demands of work, play, and other activities require quite a bit of high school algebra and geometry? This question can be the basis of project-based learning in which students interview adults and individually or jointly compile the results.

Another related topic is the question of how calculators/computers, "cash" registers, and other devices with built-in computational capabilities have changed the math education needs of adults. In a typical day, what math does a typical adult actually do mentally, by using pencil and paper, or by using calculator/computer aids? Again, this is a project that can be done by individual students, but is quite well suited to teams of any size.

Still another type of project related to folk math is to explore how people did math thousands of years ago and in different parts of the world. How did merchants and customers do the math they needed to do when they did not know how to read and write? The abacus is a piece of this history.

Projects about "Why Learn This?"

As students progress upward through the elementary, middle, and high school math curriculum, many eventually ask questions similar to, "Why am I being expected to learn this?" and "What's in it for me?"

These types of questions can be the basis for ongoing, individual projects. There are two general types of answers. First, there are the assertions that, "It will be good for you" and "You will need it in the future." Second, there are the personal answers that students can develop for themselves, answers that are specifically tailored to a student's personal needs and ambitions.

A teacher can provide the "assertion" answers, and may be able to provide some general-purpose answers that tie in with the interests of some students. An example of such an answer is, "If you want to be a (teacher names several careers) when you grow up, you will need to take and pass math courses that are based on this material." Thus, for example, people majoring in engineering in college have to take calculus and other "higher" math.

The idea underlying making this into a math project-based learning assignment (possibly a project that continues all year) is to put the responsibility onto the student. Encourage the students to think about current and possible future mathematical needs each may have. Get them to analyze the math curriculum from a personal future point of view.

As an example, have each student select a possible vocation to be pursued after leaving school. The project is to explore possible math-related needs in this vocational area and work to achieve a level of expertise that fits their personal needs and/or the needs of possible employers.

The goal is to get students to explore the vocation in some depth. Will this job or career exist five to ten years from now? Are people working at this type of job happy with and fulfilled by the work?

Projects for Use in Math Education for Teachers

"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful." (Henri Poincare; French physicist and mathematician; 1854-1912.)

This section is especially aimed at people who teach preservice and inservice courses and workshops for math teachers. The theme is one of role modeling the use of project-based learning in a manner that will help their students to make immediate use of the information and material in their own learning and in the future as they work with the students they teach.

A number of topic areas are listed below. Each can be the basis for individual or small group projects designed to produce products and presentations that can be shared with current fellow students and made available to other current and future preservice and inservice teachers.

  • Math and/or math education humor.
  • Math and/or math education history. If a student is going to learn one important "tidbit" of math history each year (or, half-year, or quarter-year) in our math education system, what should these tidbits be. Why?
  • Math and/or math education quotes. Ask and answer the same underlying question as for the line just above. See the IAE-pedia document, Math education quotations.
  • Math education research or general education research that is "powerful" enough to lead to significant improvements in math education.
  • Women and/or any other under-represented population in math.
  • Math modeling and computational thinking. What are these two topics, and to what extent do they overlap? Develop some illustrations of the use of computational thinking in learning the math you are currently studying or have studied in the past.
  • Math maturity, knowledge, and skills self-assessment.
  • Roles of calculators and computers in math and/or math education. What math problems are we learning to solve in our math courses that computers cannot currently solve?

Selecting Math Projects for Students in K-12 Education

A 3/3/2016 Google search of the expression K-12 math projects produced over 30 million results. This large number of results suggests that there likely are many sites that contain examples of suitable math projects.

However, this raises the question of what are the characteristics of a suitable (good) project? You remember the statement, "Beauty is in the eye of the beholder." A somewhat similar statement holds for math projects. One approach is to develop a general set of criteria, and to judge projects against this set of criteria.

The Computing Technology for Math Excellence (CM4ME) site is titled, K-12 Math Projects: About Project-based Learning. Quoting from this site:

Projects help students personalize their learning and are ideal for gaining key knowledge and understanding of content and answering the question: Where am I ever going to use this?" Among the greatest benefits of project-based learning (PBL) are gains in students' critical-thinking skills and development of their interpersonal and intrapersonal skills. PBL is also an ideal way to help learners gain speaking and presentation skills identified in the Common Core Standards. PBL in mathematics, particularly when completed in teams, helps learners "model with mathematics" as they "apply the mathematics they know to solve problems arising in everyday life, society, and the workplace," "use tools strategically," and "construct viable arguments and critique the reasoning of others," as noted in the Common Core Standards (2010) for Mathematical Practice.
However, as Bryan Goodwin (2010) found in reviewing the literature, a major shortcoming in many student projects is that educators tend to assign projects just for the sake of doing them. "Educators can avoid this phenomenon and realize the potential of projects to promote students' critical-thinking by framing projects around a driving question"

This article continues by raising the following list of questions to be used in selecting suitable math projects:

  1. Is the project devoted only to mathematics (or a single subject area), or is there a link to other curricular areas?
  2. Is the project tied to standards for the curricular areas addressed, such as those from the National Council of Teachers of Mathematics, the Common Core Standards, or the National Education Technology Standards?
  3. Does the project come with classroom instructional materials (e.g., teacher resources, student activities, rubrics and assessment tools)?
  4. Can all students in your class participate? Projects should not be reserved for your talented and gifted students, as all students should be able to benefit.
  5. What is the total time for project completion?
  6. Is the project collaborative in nature? A collaborative project, particularly involving students outside your own school setting, will take more time and monitoring to help students learn how to be a part of a team and communicate appropriately with others.
  7. How will students benefit both academically and personally from their involvement in the project? Consider that when students interact with other students and experts across the country or internationally, they get a broader feel for diversity. Their participation in an actual real world activity might encourage them to do their best work, and see the relevance of mathematics in their daily lives. If students have input into project selection, and like the topic, they will tend to become more involved and excited about their learning.
  8. Is there a cost involved to participate?

Some of these questions need to be answered specifically by a teacher who is considering use of a particular project, and others can be answered by people publishing lists of math projects. For example, an article may claim or suggest that a particular project is suitable for use by fifth graders. However, the fifth grade math curriculum and fifth grade students vary considerably throughout the country. Math teachers need to examine a project in terms of the math abilities and math project background and experience of their students.

Another quite important question that is not raised in the above list is whether the teacher who wants to use a particular math project has the teaching and class management skills, the math content knowledge, and the non-math content knowledge that is required.

Additional Resources

Blumenfeld, P., Soloway, E., Marx, R., Krajcik, J., Guzdial, M., & Palincsar, A. (1991). Motivating project-based learning: Sustaining the doing, supporting the learning. Retrieved 3/2/2016 from https://www.researchgate.net/file.PostFileLoader.html?id=5677ad735cd9e3849c8b4577&assetKey=AS%3A309004255858690%401450683763438. Quoting the Abstract of this seminal article:

Project-based learning is a comprehensive approach to classroom teaching and learning that is designed to engage students in investigation of authentic problems. In this article, we present an argument for why projects have the potential to help people learn; indicate difficulties that students and teachers may encounter with projects; and describe how technology can support students and teachers as they work on projects, so that motivation and thought are sustained.

Boss, S. (n.d.) Ten top tips for assessing project-based learning. Edutopia. Retrieved 3/2/2016 from http://www.edutopia.org/node/447923/guide-download?key=1456973430&token=afb56613ee18d98c77b593f765d10494. Here is a list of the tips that are discussed:

  1. Keep It Real with Authentic Products
  2. Don’t Overlook Soft Skills
  3. Learn from Big Thinkers
  4. Use Formative Strategies to Keep Projects on Track
  5. Gather Feedback—Fast
  6. Focus on Teamwork
  7. Track Progress with Digital Tools
  8. Grow Your Audience
  9. Do-It-Yourself Professional Development
  10. Assess Better Together

Miller, A. (2/28/2011). Criteria for effective assessment in project-based learning. Edutopia. Retrieved 3/2/2016 from http://www.edutopia.org/blog/effective-assessment-project-based-learning-andrew-miller. Quoting from the article:

One of the greatest potentials for PBL is that it calls for authentic assessment. In a well-designed PBL project, the culminating product is presented publicly for a real audience. PBL is also standards-based pedagogy. Oftentimes when I consult and coach teachers in PBL, they ask about the assessment of standards. With the pressures of high stakes testing and traditional assessments, teachers and administrators need to make sure they accurately design projects that target the standards they need students to know and be able to do.
In addition, teachers need to make sure they are continually assessing throughout a PBL project to ensure their students are getting the content knowledge and skills that they need to complete the project. Below are some criteria to ensure that your PBL project demands that demands high expectations, aligned to standards and assessed properly.

Miller, A. (5/10/2011). Tips for using project-based learning to teach math standards. Edutopia. Retrieved 3/2/2016 from http://www.edutopia.org/blog/project-based-learning-math-standards. Quoting from this article:

I know the structures in place for Math teachers. Sometimes there is not enough time for a project. Sometimes, it's just not the best use of time. If a standard needs to be covered in a short week unit, then it isn't the best place for a project. However, if there is a 3-week unit coming up around a specific math learning target, this would be a great opportunity to create a project. There is time and space for you the teacher to get your "feet wet" in implementing the project. In addition, you might be able to combine the learning targets in a project that seem to fit together. Your allow time increases and you can have students create products that demonstrate learning of both targets or standards. As a teacher, be creative with the time you have, either in looking for the best opportunity or creating an opportunity.

NMSA (n.d.). Project-based learning in middle grades mathematics. National Middle School Association. Retrieved 3/2/2016 from http://webcache.googleusercontent.com/search?q=cache:5ji1bm4fR0AJ:aggiestem.tamu.edu/sites/aggiestem.tamu.edu/files/resources/PBL%2520mathematics%2520research%2520summary_0.pdf+&cd=1&hl=en&ct=clnk&gl=us. Quoting from the document:

The focus of this research summary is to foster an understanding of project-based learning (PBL), particularly in mathematics education; to explain the factors for making a conscious decision to implement PBL in middle grades mathematics classrooms; and to provide insights about the possible realized effects when mathematics-based PBL is implemented.
The teacher’s belief system is paramount. A teacher who believes that social constructivism (Vygotsky, 1978) or situated learning (Boaler, 1999; Cobb, 1988) is useless, will find the work and effort for accomplishing PBL to outweigh its benefits. The tenets of constructivism, in its many versions, underlie PBL designs (Grant, 2002). In PBL, the teachers’ role necessitates that they allow all students to engage in developing personally and collaboratively negotiated meanings from the learning event (Harel & Papert, 1991; Kafai & Resnick, 1996). The success of PBL should be assessed on many levels including emotional development, collaboration, leadership, and negotiation skills that are essential for project success (Glaser, 1992; Glennan, 1998). When teachers allow students more autonomy over what they learn, it improves motivation, and students assume more responsibility for their learning (Tassinari, 1996; Wolk, 1994; Worthy, 2000). However, this does not mitigate the important need for the teacher to be actively engaged in the learning task as both a role model and an advisor. This role also does not forsake whole-group didactic instruction, but makes careful use of it to address learning deficits. The teacher can function as a co-constructor of knowledge (Rosenfeld & Rosenfeld, 2006). In this role, the teacher must possess profound content knowledge, be confident in his or her skill to facilitate learners of diverse abilities, and be prepared to deal with a more diverse set of questions—potentially across disciplines. Consequently, the role of the teacher and this diversity in content raise questions about the scope and style of assessing student learning.

Author or Authors

The original version of this document was developed by David Moursund. A number of people—and especially Bob Albrecht—have suggested ideas and materials for inclusion in this document.